Abstract
With the increasing application of (probabilistic) hesitant fuzzy sets in decision-making, the existing integrated methods of hesitant fuzzy information have become too complicated to meet the needs of increasingly complex practical decision-making problems. Therefore, this paper combines the related knowledge of probability theory to firstly introduce the concept of continuous hesitant fuzzy element (C-HFE). Subsequently, the concept of uniform hesitant fuzzy element (U-HFE) is proposed, and discrete (probabilistic) hesitant fuzzy information is transferred to continuous one, benefited from the connection between U-HFEs and C-HFEs with uniform distribution. After then, integration methods of C-HFEs based on mathematical derivation are developed, which lays a theoretical foundation for the continuity of hesitant fuzzy information. Further, facing the problem that the method of mathematical derivation is too tedious, based on computer simulation, this paper proposes another integration method of C-HFEs, which is more concise and easier to apply. Finally, an example of the evaluation of water resources emergency management plans is given to apply the above method to practical decision-making problems.
Highlights
Since the advent of fuzzy sets in 1965 [1], various extended forms have been proposed [2,3,4,5,6]
We develop an integration method of continuous hesitant fuzzy element (C-hesitant fuzzy element (HFE)) with normal distribution and uniform distribution based on mathematical derivation, which lays a theoretical foundation for the operation and integration of C-HFEs
The main work of this paper is to try to use the concept of density function to process the originally discrete hesitant fuzzy information continuously based on the combining of probability theory and fuzzy decision-making theory
Summary
Since the advent of fuzzy sets in 1965 [1], various extended forms have been proposed [2,3,4,5,6]. Simpler calculation methods are needed to deal with increasingly complex practical decision-making (especially large-scale group decision-making) problems To address such an issue, some scholars have proposed their own solutions from different perspectives [20,21]. We proposed the concept of uniform hesitant fuzzy element (UHFE) and used its connection with C-HFE with uniform distribution to link discrete HFEs with C-HFEs. we develop an integration method of C-HFEs with normal distribution and uniform distribution based on mathematical derivation, which lays a theoretical foundation for the operation and integration of C-HFEs. After that, a simpler aggregation method of C-HFEs based on computer simulation is introduced to make the C-HFEs easier and more effective when applying to solve decision-making problems.
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